Bayesian Analysis of Flipper Length

Chinstrap Penguins

Heather Anderson, Maria Drury, Hailee Hawkins, Carolyn Herrera, Liz Miller, Sara Parrish (Advisor: Dr. Seals)

Nov 23, 2024

The Species: Chinstrap Penguins

  • Chinstrap penguins are one of the most abundant penguin species in the Antarctic, with an estimated population of nearly eight million pairs. They are the most abundant penguin species breeding at the South Sandwich Islands. According to the study, the penguins take more than 10,000 naps a day, with each one lasting only four seconds, totaling 11-12 hours per day.
  • Today we will analyze flipper lengths for Chinstrap penguins.

Image Source: Gemini AI

Bayesian Model Selection

  • Since flipper lengths are generally well-distributed and approximately normal for penguins, the Normal-Normal Bayesian Model is the most appropriate for the analysis of Chinstrap penguin flipper lengths.

Y_i|\mu \sim N(\mu, \sigma^2)

Prior Specification: The Data

  • Using prior studies we have determined the average flipper length of Chinstrap penguins is 193.37mm with a standard deviation of 6.12 [1]

Image Source: wikipedia.org

Prior Specification: The Model

\mu \sim N(193.7, {6.12}^2)

The Sample Data

  • There are 68 data points for the Chinstrap penguins. The sample mean is 195.8235mm with a standard deviation of 7.131894.

The Posterior Model

\mu|(Y = 68) \sim N(?,?) \ \sigma = 7.13, \ n = 68, \ \tau = 6.12, \ \bar{y} = 193.37

\mu | \overset{\to}y \sim N\left( 195.82 \frac{7.13 ^2}{68*6.12^2 + 7.13^2} + 193.37\frac{68*6.12^2}{68*6.12^2 + 7.13^2}, \frac{6.12^2 7^2}{68*6.12^2 + 7.13^2} \right)\\

\text{Simplified: N} \left( 195.82 * 0.01957 + 193.37 * 0.9804, 0.7329 \right)\\

\text{N: } (193.42, 0.856^2)

Model Mean Mode Variance SD
Prior 193.37 193.37 37.4544 6.12
Posterior 195.7721 195.7721 0.7329711 0.8561373

Credible Intervals

qnorm(c(0.025, 0.975), 195.77, 0.86)
[1] 194.0844 197.4556

95\% \text{ CI} = [194.08, 197.46]

Image Source: commons.wikimedia.org

Hypotheses

Chinstraps: 215 and 225 mm

  • Null Hypothesis H_0:

    • H_0: \mu \in (215,225) 

  • Alternative Hypothesis H_1:

    • H_1: \mu \notin (215,225)

Image Source: Gemini AI

Hypotheses: Conclusions

  • The 95% credible interval for the posterior is [194.08, 197.46], which does not overlap with the range specified in the null hypothesis [215, 225]. This indicates the posterior belief supports values below 215mm, resulting in no evidence for the null hypothesis.

Image Source: crittersquad.com

Posterior Probability

posterior_prob <- pnorm(225, 195.77,0.86)-pnorm(215,195.77,0.86)
posterior_prob
[1] 0
posterior_odds <- (posterior_prob/(1-posterior_prob))
posterior_odds
[1] 0

P(215\le\mu\le225|Y=68)=0

This confirms the conclusion drawn previously.

Image Source: redbubble.com

Bayes Factor

prior_prob <- pnorm(225,193.37, 6.12)-pnorm(215,193.37,6.12)
prior_prob
[1] 0.0002042996
prior_odds <-(prior_prob/(1-prior_prob))
prior_odds
[1] 0.0002043414
bayes_factor <- posterior_odds/prior_odds
bayes_factor
[1] 0

\text{Bayes Factor}= \frac{\text{Posterior Odds}}{\text{Prior Odds}}= \frac{0}{0.0002}=0

  • Since the Bayes Factor is less than 1, the chance of the average flipper length being between 215mm and 225mm decreased in light of the data. This along with the 95% credible interval confirms our belief that the average flipper length is not between 215mm and 225mm.
Hypotheses Prior Probability Posterior Probability
H_0: \mu \in (215,225) P[H_0] = 0.9998 P[H_0 \mid Y = 68] = 1
H_1: \mu \notin (215,225) P[H_1] = 0.0002 P[H_1 \mid Y = 68] = 0

Summary

  • The average posterior flipper length for Chinstrap penguins is 195.77 with a standard deviation of 0.86, while our average prior flipper length for Chinstrap penguins is 193.37 with a standard deviation of 6.12. We noted no extra large penguins in our sample.

Image Source: Gemini AI

References

[1]
Moreno, Barbosa, León, and Fargallo, “Phenotypic selection on morphology at independence in the chinstrap penguin pygoscelis antarctica,” J. Evol. Biol., vol. 12, no. 3, pp. 507–513, May 1999.